To solve this problem, we need to determine how many possible passwords can be formed with the following conditions:

• The password is 8 characters long.
• The first 2 characters are letters.
• The last 6 characters are digits.
• No repetition of any letter or digit is allowed.

Step-by-step breakdown:

1. Choosing the letters:

   • There are 26 letters in the English alphabet.
   • For the first letter, we can choose any of the 26 letters.
   • For the second letter, since repetition is not allowed, we can choose from the remaining 25 letters.

   So, the number of ways to choose the letters is: $26 \times 25$

2. Choosing the digits:

   • There are 10 possible digits (0 through 9).
   • For the first digit, we can choose any of the 10 digits.
   • For the second digit, we can choose from the remaining 9 digits.
   • For the third digit, we can choose from the remaining 8 digits, and so on, until the sixth digit, where we have 5 remaining digits.

   So, the number of ways to choose the digits is: $10 \times 9 \times 8 \times 7 \times 6 \times 5$

Total number of passwords:

The total number of possible passwords is the product of the number of ways to choose the letters and the number of ways to choose the digits: $(26 \times 25) \times (10 \times 9 \times 8 \times 7 \times 6 \times 5)$

Let's calculate this:

$26 \times 25 = 650$

$10 \times 9 \times 8 \times 7 \times 6 \times 5 = 1512000$

So, the total number of possible passwords is: $650 \times 1512000 = 983400000$

Final answer:

There are 983,400,000 possible passwords.

Would you like further clarification or have any questions? Here are a few related questions you might find interesting:

1. How would the number of possible passwords change if repetition of letters and digits were allowed?
2. How many possible passwords are there if the password is 10 characters long, with 4 letters and 6 digits, with no repetition?
3. How do we calculate the total number of possible arrangements if there are specific constraints like requiring alternating letters and digits?
4. What is the probability of guessing a password correctly on the first try?
5. How can we apply similar counting techniques to more complex password requirements?

Tip: For problems involving permutations like this, it's helpful to break the problem into manageable parts: first choosing letters, then digits, and finally multiplying the possibilities.